Arithmetic Progression: The nth term

To find the nth term is like taking a shortcut through a fixed sequence without having to work out each step. Using this formula a+(n -1)d you can jump to any part of a recurring sequence to find the values.

To understand the formula:

a= the amount starting in the sequence
d= the amount added at each position of the sequence or the common difference
n= the position of any term in the sequence

Term	n
1	1	a
2	2	(a)+d
3	3	(a+d)+d or a+2d
4	4	(a+2d)+d or a+3d
5	5	(a+3d)+d or a+4d

Practical Example

Lets try to get to the 15th term in this sequence; 5,9,13,17

So a=5 as it's the starting value
d= 4, the common difference between terms

Using the formula: a+(n+1)d

Step 1: 5+(15-1)4 (insert values)

Step 2: 5+(14x4) = 61

Next we will try to find the sum of a specific sequence length, for this example we want to know the sum of n20 in the sequence above. The formula to work this out has one missing value "l", which is the last value in the sequence upto n20. We will need to find this out before we can work out the sum of terms upto n20.

The formula to work out l is l=a+(n-1)d
so lets plug in the values so we can get to l.

l= 5+(20-1)4
Therfore; l=81

Now we have l we can work out the sum of all terms(Sn20) up untill n20 with the following formula...

So with values inserted...

Sn = 860